Quasi-Cuspidal Representations and Generators of Equivariant D-Modules

Definitions and mathematical statement

Let $G$ be a connected reductive group and let $\mathfrak g=\operatorname{Lie}(G)$. Write $U\mathfrak g$ for its universal enveloping algebra and $D_{\mathfrak g}$ for the intended algebra of differential operators on $\mathfrak g$. For a representation $V$, regarded as a left $U\mathfrak g$-module, the source sets \[ P(V)=D_{\mathfrak g}\otimes_{U\mathfrak g}V. \] This tensor product requires a right $U\mathfrak g$-module structure on $D_{\mathfrak g}$, which the supplied record does not specify. Denote the source's intended category of $G$-equivariant $D$-modules on $\mathfrak g$ by $D\!\operatorname{-mod}(\mathfrak g)^G$.

Subject to fixing the missing conventions, the source asks the following questions for a connected reductive $G$:

1. Characterize the representations $V$ described as quasi-cuspidal for which $P(V)$ has a nonzero cuspidal direct summand; explicitly, for which there is a decomposition \[ P(V)\cong C\oplus M, \qquad C\ne 0, \] with $C$ cuspidal in the intended category.
2. Find a minimal set $\mathcal S$ of irreducible representations $V$ such that the objects $P(V)$, for $V\in\mathcal S$, generate $D\!\operatorname{-mod}(\mathfrak g)^G$ in the intended sense.

The supplied record does not define quasi-cuspidality or cuspidality, specify the base field or its characteristic, identify the permitted representations, or distinguish an abelian category from a derived category. It also leaves the generation operation and the meaning of minimality unspecified. Consequently, these questions retain the source's targets but require additional conventions before they constitute a fully determined mathematical problem. The supplied background additionally mentions minimal generators of generalized Springer blocks, without specifying the blocks or their relationship to the whole-category generation question.

Short English statement

For a connected reductive group $G$, characterize the quasi-cuspidal representations whose induced modules $P(V)$ contain a cuspidal direct summand, and find a minimal collection of irreducible representations whose $P(V)$ generate the indicated equivariant $D$-module category. Several essential definitions and conventions are missing from the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004019$, exact source problem code catalogue entry, titled Quasi-Cuspidal Representations and Generators of Equivariant D-Modules, in the collection Oberwolfach Reports Open Problems. The registry records proposed year 2018 and supplies no individual proposer attribution.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2019), Oberwolfach Reports, pp. 3081--3146, DOI 10.4171/owr/2018/52. https://doi.org/10.4171/owr/2018/52. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 52/2018. https://ems.press/content/serial-article-files/46773.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the report provides examples and partial structure, rather than a complete characterization of quasi-cuspidal representations or minimal generators of generalized Springer blocks. This is supplied dated curation, not an independently verified literature assessment. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired. This is a supplied editorial label, not independent mathematical verification of this draft.